### 3.407 $$\int e^{4 \coth ^{-1}(a x)} (c-\frac{c}{a x})^2 \, dx$$

Optimal. Leaf size=27 $-\frac{c^2}{a^2 x}+\frac{2 c^2 \log (x)}{a}+c^2 x$

[Out]

-(c^2/(a^2*x)) + c^2*x + (2*c^2*Log[x])/a

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Rubi [A]  time = 0.121187, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.182, Rules used = {6167, 6131, 6129, 43} $-\frac{c^2}{a^2 x}+\frac{2 c^2 \log (x)}{a}+c^2 x$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(4*ArcCoth[a*x])*(c - c/(a*x))^2,x]

[Out]

-(c^2/(a^2*x)) + c^2*x + (2*c^2*Log[x])/a

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rule 6131

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> Dist[d^p, Int[(u*(1 + (c*x)/d)
^p*E^(n*ArcTanh[a*x]))/x^p, x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 0] && IntegerQ[p]

Rule 6129

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^p, Int[(u*(1 + (d*x)/c)
^p*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ
[p] || GtQ[c, 0])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int e^{4 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^2 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^2 \, dx\\ &=\frac{c^2 \int \frac{e^{4 \tanh ^{-1}(a x)} (1-a x)^2}{x^2} \, dx}{a^2}\\ &=\frac{c^2 \int \frac{(1+a x)^2}{x^2} \, dx}{a^2}\\ &=\frac{c^2 \int \left (a^2+\frac{1}{x^2}+\frac{2 a}{x}\right ) \, dx}{a^2}\\ &=-\frac{c^2}{a^2 x}+c^2 x+\frac{2 c^2 \log (x)}{a}\\ \end{align*}

Mathematica [A]  time = 0.103305, size = 29, normalized size = 1.07 $-\frac{c^2}{a^2 x}+\frac{2 c^2 \log (a x)}{a}+c^2 x$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(4*ArcCoth[a*x])*(c - c/(a*x))^2,x]

[Out]

-(c^2/(a^2*x)) + c^2*x + (2*c^2*Log[a*x])/a

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Maple [A]  time = 0.045, size = 28, normalized size = 1. \begin{align*} -{\frac{{c}^{2}}{{a}^{2}x}}+x{c}^{2}+2\,{\frac{{c}^{2}\ln \left ( x \right ) }{a}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x-1)^2*(a*x+1)^2*(c-c/a/x)^2,x)

[Out]

-c^2/a^2/x+x*c^2+2*c^2*ln(x)/a

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Maxima [A]  time = 1.0413, size = 36, normalized size = 1.33 \begin{align*} c^{2} x + \frac{2 \, c^{2} \log \left (x\right )}{a} - \frac{c^{2}}{a^{2} x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2*(c-c/a/x)^2,x, algorithm="maxima")

[Out]

c^2*x + 2*c^2*log(x)/a - c^2/(a^2*x)

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Fricas [A]  time = 1.56775, size = 65, normalized size = 2.41 \begin{align*} \frac{a^{2} c^{2} x^{2} + 2 \, a c^{2} x \log \left (x\right ) - c^{2}}{a^{2} x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2*(c-c/a/x)^2,x, algorithm="fricas")

[Out]

(a^2*c^2*x^2 + 2*a*c^2*x*log(x) - c^2)/(a^2*x)

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Sympy [A]  time = 0.292655, size = 26, normalized size = 0.96 \begin{align*} \frac{a^{2} c^{2} x + 2 a c^{2} \log{\left (x \right )} - \frac{c^{2}}{x}}{a^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)**2*(a*x+1)**2*(c-c/a/x)**2,x)

[Out]

(a**2*c**2*x + 2*a*c**2*log(x) - c**2/x)/a**2

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Giac [B]  time = 1.12239, size = 127, normalized size = 4.7 \begin{align*} -\frac{2 \, c^{2} \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a} + \frac{2 \, c^{2} \log \left ({\left | -\frac{1}{a x - 1} - 1 \right |}\right )}{a} + \frac{c^{2} + \frac{2 \, c^{2}}{a x - 1}}{a^{2}{\left (\frac{1}{{\left (a x - 1\right )} a} + \frac{1}{{\left (a x - 1\right )}^{2} a}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2*(c-c/a/x)^2,x, algorithm="giac")

[Out]

-2*c^2*log(abs(a*x - 1)/((a*x - 1)^2*abs(a)))/a + 2*c^2*log(abs(-1/(a*x - 1) - 1))/a + (c^2 + 2*c^2/(a*x - 1))
/(a^2*(1/((a*x - 1)*a) + 1/((a*x - 1)^2*a)))